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Akshay Rao, marketing professor

Marketing expert Akshay Rao has found that smart people often miscalculate when dealing with percentages.

Pitfalls of percentages

A new study warns people to be on their toes when dealing with percentage changes

By Deane Morrison

May 25, 2007

Several years ago, a nurse at a Twin Cities hospital temporarily filled in for a head nurse who took a short leave. The nurse was given a 10 percent raise; when she returned to her old job, she took a 10 percent pay cut. If that sounds equitable to you, Akshay Rao, General Mills Professor of Marketing at the Carlson School of Management, wishes to point out a few things. Rao and Haipeng Chen, a Carlson school Ph.D. who is now an assistant professor at the university of Miami, have completed a study showing that people often treat percentages like whole numbers, resulting in systematic calcuation errors. Their paper will be published in an upcoming issue of the Journal of Consumer Research. The nurse is a perfect example of the kind of error they're talking about. Suppose her original salary was $1,000 a week. A 10 percent raise would bump it up by $100, to $1,100. But 10 percent of that salary is $110, not $100. Subtracting $110 from $1,100 would leave her with only $990--a one percent cut from her original salary. "It took a month to convince [the human resources department] that she now earned less than before," says her husband. The error was corrected. In their paper, Rao and Chen report that a common problem with percentages arises when consumers are unaware of how multiple discounts work. For example, if a store offers 25 percent off everything but with "an extra 25 percent" off certain items, that doesn't mean 50 percent off the original price. Instead, the actual discount amounts to about 43 percent. That's because the store will take the second 25 percent off the discounted price. Therefore, on a $100 item, the second discount will be 25 percent of $75, for a final price of $56.25--not $50. Firms can benefit at the expense of consumers who don't realize this. In their study, Rao and Chen tested the impact of offering a 20 percent discount and an additional 25 percent discount versus an equivalent 40 percent discount in a retail store. When double discounts were offered, the store reported more buyers, sales volume, revenue, and profit. In the case of double discounts, it's understandable that consumers would not know how the second discount is computed. After all, the two discounts are applied only once, at the cash register.

"Imagine your stock portfolio went up 40 percent last period, and down 30 percent this period," Rao says. "You are not better off by 10 percent. Your portfolio is down two percent."

But in other cases, such as fluctuations in the stock market, time elapses between one percentage change and the next. Even then, some people forget that they must continually adjust base values when computing sequential percentage changes. When a value goes up and then down (or vice versa), things can really get interesting.

The Times slipping away

Even experts make mistakes. In their paper, Rao and Chen offer this gem of miscalculation:

"The depression took a stiff wallop on the chin here today. Plumbers, plasterers, carpenters, painters and others affiliated with the Indianapolis Building Trades Unions were given a 5 percent increase in wages. That gave back to the men one-fourth of the 20 percent cut they took last winter." The New York Times, quoted in How to Lie with Statistics (Darrell Huff 1954, p. 111).

"Imagine your stock portfolio went up 40 percent last period, and down 30 percent this period," Rao says. "You are not better off by 10 percent. Your portfolio is down two percent." The problem with percentages isn't that we're not smart; it's just that percentages are a relatively new mental exercise. But with effort, we can compensate. "We argue for increased math education because there are some calculations that aren't innate to the mammalian brain," Rao explains. "For example, the brain can detect changes in light intensity. This is a very sophisticated calculation for the brain, but it confers an evolutionary advantage. For instance, a change in light intensity may mean a predator is approaching. But percentages are an artificial construct that requires learning. From a public policy standpoint, we have to teach people how to do this." Sometimes, mistakes can have national ramifications. Rao gives the hypothetical case of the Pentagon asking for the same percentage increase in its budget every year. But as a budget grows, so does the number of dollars represented by that percentage. If Congress tried to increase the appropriation by the same dollar amount rather than the same percentage, the story could be spun as Congress cutting the military budget, says Rao. Or, suppose Detroit improves the mileage of its fleet by, say, two miles per gallon each year. The percentage improvement will be less and less, even though the rate of progress toward a mileage goal remains the same. Rao's advice? "Make sure you know what the base of calculation is, especially for multiple percentage changes," he says. And if you're shopping for sales, bring a calculator.